203.-PROBLEMS.

1. Wanting to know the distance between two inaccessible objects, which lie in a direct level line from the bottom of a tower 120 feet in height, the angles of depression are measured from the top of the tower, and are found to be, of the nearer 57°, and of the more remote 25° 30'; required the distance between the objects. Ans. 173.656 feet.

2. In order to find the distance between two trees, A and B, which could not be directly measured because of a pool which occupied the intermediate space, the distances of a third point C from each of them were measured, and also the included angle ACB; it was found that,

FIG. 99.

3. Being on a horizontal plane, and wanting to ascertain the height of a tower standing on the top of an inaccessible hill, there were measured, the angle of elevation of the top of the hill 40°, and of the top of the tower 51°; then measuring in a direct line 180 feet farther from the hill, the angle of elevation of the top of the tower was 33° 45': required the height of the tower.

Ans. 83.998.

4. Wanting to know the horizontal distance between two inaccessible objects E and W, the following measurements were made;

other equal to 200 yards; from the former of these points A could be seen, and from the latter B, and at each of the points C and D a staff was set up. From C a distance CF was measured, not in the direction DC, equal to 200 yards, and from Da distance DE equal to 200 yards, and the following angles taken,

(AFC 83° 00′, BDE = 54° 30',

6. From a station P there can be seen three objects, A, B, and C, whose distances from each other are known: viz.,

=

AB 800, AC 600, and

and BC 400 yards.

Now, there are measured the horizontal angles,

APC 33° 45', and BPC = 22° 30';

is required to find the three distances, PA, PC, and PB.

GEOMETRICALLY.

With the three given sides construct the triangle ABC. Then, at A lay off the angle BAD 22° 30', and at B the angle ABD 33° 45', and note D, the point at which the two lines intersect.

Through the points A, D, and B, describe the circumference of a circle, and through C and D draw the line CDP; the point P in which it intersects the circumference, will be the position of the station.

FIG. 102.

B

By observing the equal angles in the figure, the trigonometrical solution is not difficult.

Ans.

We find,

NOTE.-This problem is much used in maritime surveying, for the purpose of locating buoys and sounding-boats. The trigonometrical solution is somewhat tedious, but the geometrical solution is very easy, as shown above.

SECTION III.

RANGING OUT LINES, ETC.

204. To range out a line with the transit, place the instrument, carefully adjusted, over the first station; direct the telescope to a distant and well-defined point in the desired line, and clamp both the vernier plate and the horizontal limb. The line of sight of the telescope is then in the vertical plane of the given line, so that points on the surface bisected by the intersection of the cross-wires will be in the required line; let an assistant, directed into the line by the observer at the transit, fix ranging-rods, or stakes conspicuously marked, as far as the power of the telescope extends. Remove the transit to the third or fourth stake from the last set, and place it precisely over that position by plumb-bob, and adjust it for observation; the telescope is ranged in the line by sighting backwards and forwards to the stakes already set. The line is then continued

as before.

If great accuracy be required, each operation must be repeated with telescope reversed, as only in this way can error in adjustment of cross-wires be eliminated. If the sighting with reversed telescope does not agree with the former sighting, take a point midway between the two points sighted, as a point in the required line.

205. A line may be traced in forests or plantings, in which there are no general surface obstructions, by the aid of auxiliary parallel lines.

In the illustration, Fig. 103, aa, bb, cc, are the parallel lines, and aa, cc, the auxiliary lines. AB is a line in which b is a given point. The distances ab, bc, in this line should be measured,

and also the angle which bb makes with AB. The line bb and the auxiliary lines should be traced on this angle, until the trace of one of the parallel lines be obstructed by a tree,

such as bb at (1).

Immediately on passing the obstruction on one of the auxiliary lines, a line should be traced on the measured angle or its supplement, as may be required, and traced to intersect the other auxiliary line. The angle made by these lines should be measured at the point of intersection to verify the trace of the intersecting lines. From these angular points in the auxiliary lines, distances to the point (1), equal to ab and cb, respectively, should be measured in the transverse line, and found to meet, but not overlie, one another. Then will the point of meeting in the transverse line be a forward point in the line bb. At a suitable distance forward from which the point (1) may be observed, a like determination of another point in bb should be made. The trace of the line bb should be taken upon these points and continued in connection with the auxiliary lines until the trace of one of the lines be obstructed, such as the line aa at (2). The trace of the obstructed line should be taken up by measurements in the transverse line